Pseudo-arc

A hereditarily-indecomposable snake-like continuum which contains more than one point.

Comments

One speaks of "the" pseudo-arc, since any two are homeomorphic [a2]. Like the arc $[0,1]$, the pseudo-arc is homeomorphic to each of its non-degenerate subcontinua [a7]. Yet, like the circle, it is homogeneous [a1]. Its unique feature — necessarily unique — is that "almost-all continua are pseudo-arcs" ; more precisely, in the hyperspace of subcontinua of an $n$-cell for $n\geq2$, the pseudo-arcs form a residual set [a3]. All non-degenerate homogeneous snake-like continua are pseudo-arcs [a4]. Simpler proofs of the fundamental properties, and some generalizations, are developed in [a5], [a6], [a8].

References

[a1]	R.H. Bing, "A homogeneous indecomposable plane continuum" Duke Math. J. , 15 (1948) pp. 729–742
[a2]	R.H. Bing, "On snake-like continua" Duke Math. J. , 18 (1951) pp. 853–863
[a3]	R.H. Bing, "Concerning hereditarily indecomposable continua" Pacific J. Math. , 1 (1951) pp. 43–51
[a4]	R.H. Bing, "Each homogeneous nondegenerate chainable continuum is a pseudo-arc" Proc. Amer. Math. Soc. , 10 (1959) pp. 345–346
[a5]	J. Krasinkiewicz, "Mapping properties of hereditarily indecomposable continua" Houston J. Math. , 8 (1982) pp. 507–516
[a6]	J. Krasinkiewicz, P. Minc, "Mappings onto indecomposable continua" Bull. Acad. Polon. Sci. , 25 (1977) pp. 675–680
[a7]	E.E. Moïse, "An indecomposable plane continuum which is homeomorphic to each of its non-degenerate subcontinua" Trans. Amer. Math. Soc. , 63 (1948) pp. 581–594
[a8]	L. Oversteegen, E. Tymchatyn, "On hereditarily indecomposable compacta" H. Toruńczyk (ed.) S. Jackowski (ed.) S. Spiez (ed.) , Geometric & Algebraic Topology , Banach Center Publ. , 18 , PWN (1986) pp. 407–417